Answer : Given: A, B and C three sets are given. Need to prove: A × (B ???? C) = (A × B) ???? (A × C)
Let us consider, (x, y) A × (B ???? C)
⇒ x A and y (B ???? C)
⇒ x A and (y B or y C)
⇒ (x A and y B) or (x A and y C)
⇒ (x, y) (A × B) or (x, y) (A × C)
⇒ (x, y) (A × B) ???? (A × C)From this we can conclude that,
⇒ A × (B ???? C) ⊆ (A × B) ???? (A × C)—- (1)
Let us consider again, (a, b) (A × B) ???? (A × C)
⇒ (a, b) (A × B) or (a, b) (A × C)
⇒ (a A and b B) or (a A and b C)
⇒ a A and (b B or b C)
⇒ a A and b (B ???? C)
⇒ (a, b) A × (B ???? C)
From this, we can conclude that,
⇒ (A × B) ???? (A × C) ⊆ A × (B ???? C)—- (2)
Now by the definition of the set we can say that, from (1) and (2), A × (B ???? C) = (A × B) ???? (A × C) [Proved]